moving the armature along with the model
Simply move the armature object, or the root bone (in pose mode).



Once you have rigged a character, you no longer need to move the mesh objects themselves (usually characters consist of several mesh objects). That would be very difficult to animate, and could not deform.

This is in fact the purpose of rigging - to allow you to easily control and animate a character using bones rather than the original objects themselves.

Usually a rig will have a Master or Root bone that is used to position, rotate and possibly scale the character. This bone is the final parent of all the bones in the rig (the first bone of each chain, if not parented to something else, is parented to the root)